Combine the boiling water and seven-grain cereal in a large bowl and let stand until the grains are softened and the mixture has cooled to about 100 degrees, about 20 minutes.

Stir the olive oil into the grain mixture, then add the bread flour, whole-wheat flour, yeast, and salt. Knead the dough—either in a stand mixer with a dough hook or by hand on a floured surface—until it is smooth and elastic; if the dough remains too sticky, add the remaining bread flour a tablespoon at a time.

Place the dough in a lightly greased bowl, cover tightly with plastic wrap, and let rise in a warm, draft-free spot until doubled in size, 1 to 1.5 hours.

Adjust an oven rack to the lower-middle position, place a baking stone or inverted baking sheet on the rack, and heat the oven to 450 degrees. Toss the cherry tomatoes, zucchini, shallots, garlic, olive oil, balsamic vinegar, sugar, red pepper flakes, salt, and pepper together on a rimmed baking sheet.

Roast the vegetables until they are softened and starting to brown, about 25 minutes. Drain the mixture through a fine-mesh strainer to remove excess liquid, then increase the oven temperature to 500 degrees.

Divide the dough into three equal pieces. On a large piece of parchment paper, roll one piece of dough into a 16 by 8-inch rectangle. Repeat with the other pieces of dough on separate sheets of parchment.

Brush the edges of each dough rectangle with the remaining olive oil. Spread the roasted vegetable mixture and the feta evenly over the three doughs, leaving a 1/2-inch border around the edges.

Slide the parchment with the first flatbread onto the preheated stone. Bake until the crust is deeply browned and the cheese is soft, 8 to 12 minutes. Repeat with the remaining flatbreads.

Sprinkle the finished flatbreads with fresh basil and serve hot.